Hardy Ramanujan Journal |

Various properties of the Mellin transform function $$\mathcal{M}_k(s):= \int_1^{\infty} Z^k(x)x^{-s}\,dx$$ are investigated, where $$Z(t):=\zeta(\frac{1}{2}+it)\,\chi(\frac{1}{2}+it)^{-1/2},~~~~\zeta(s)=\chi(s)\zeta(1-s)$$ is Hardy's function. Connections with power moments of $|\zeta(\frac{1}{2}+it)|$ are established, and natural boundaries of $\mathcal{M}_k(s)$ are discussed.

Source : oai:HAL:hal-01112545v1

Volume: Volume 33

Published on: January 1, 2010

Submitted on: March 3, 2015

Keywords: analytic continuation, absolute convergence, Hardy's function, Riemann zeta-function,Mellin transforms,[MATH] Mathematics [math]

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